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Homework Help: Matrix of eigenvectors, relation to rotation matrix

  1. Apr 12, 2013 #1
    So I am given [itex]B=\begin{array}{cc} 3 & 5 \\ 5 & 3 \end{array}[/itex]. I find the eigenvalues and eigenvectors: 8, -2, and (1, 1), (1, -1), respectively. I am then told to form the matrix of normalised eigenvectors, S, and I do, then to find [itex]S^{-1}BS[/itex], which, with [itex]S = \frac{1}{\sqrt{2}}\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}[/itex], I get [itex]\begin{array}{cc} 8 & 0 \\ 0 & -2 \end{array}[/itex]. All great and dandy, that's the correct answer, and the text then asks me for the rotation matrix, which I give as [itex]\begin{array}{cc} cos\theta & -sin\theta \\ sin\theta & cos\theta \end{array}[/itex]. However, when asked how the rotation matrix and S are related, I am clearly stumped; checking the answers, they have used [itex]S = \frac{1}{\sqrt{2}}\begin{array}{cc} 1 & -1 \\ 1 & 1 \end{array}[/itex], which I think is a valid choice, since the eigenvector could have been (1,-1) or (-1,1). But since I made a different choice, I cannot get the answer they are looking for (That S is the rotation matrix for theta is 45).

    Thus, have I made a mistake in my calculation? In my reasoning? In my assumptions? Or is it just that I was unlucky to have picked the way I did and the question wasn't expecting that (I doubt the latter)?
  2. jcsd
  3. Apr 12, 2013 #2


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    You're just unlucky. The way you chose the eigenvectors, S is a rotation combined with a reflection across the line y=x. As you saw, you can choose the eigenvectors so that S is just a rotation.
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